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The basic probabilistic and statistical properties of this model are studied in depth. We proceed to derive estimators using the conditional least squares (CLS) and conditional maximum likelihood (CML) methods, while also establishing the asymptotic properties of the CLS estimator. Furthermore, we employed the CLS and CML score functions to infer the threshold parameter. Additionally, three test statistics to detect the existence of the piecewise structure and explanatory variables were utilized. To support our findings, we conducted simulation studies and applied our model to two applications concerning the daily stock trading volumes of VOW.<\/jats:p>","DOI":"10.3390\/e26020140","type":"journal-article","created":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T03:28:29Z","timestamp":1707103709000},"page":"140","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Time-Varying Mixture Integer-Valued Threshold Autoregressive Process Driven by Explanatory Variables"],"prefix":"10.3390","volume":"26","author":[{"given":"Danshu","family":"Sheng","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Liaoning University, Shenyang 110031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dehui","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Liaoning University, Shenyang 110031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3115-7740","authenticated-orcid":false,"given":"Jie","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Changchun University of Technology, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinyang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Liaoning University, Shenyang 110031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiran","family":"Zhai","sequence":"additional","affiliation":[{"name":"State Grid Jilin Electric Power Company Limited Information and Telecommunication Company, Changchun 132400, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.1080\/09603107.2010.498343","article-title":"Integer-valued moving average modelling of the number of transactions in stocks","volume":"20","year":"2010","journal-title":"Appl. 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